For this purpose, we need to adapt (REF ) and (REF ), derived in the quantum context, to the classical TBA. As stressed in [1]}, even if the TBA was originally developed to tackle quantum integrable systems, its general structure is not limited to those: classical limits have been investigated in [2]} or [3]}, and [4]} explored TBA approaches to soliton-bearing systems. In all cases, the TBA structure remains valid, with \(F(\epsilon )\)  being replaced by the free energy function representing the correct statistics of the asymptotic particles.
